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Restriction of toral
eigenfunctions to totally geodesic submanifolds
Xiaoqi Huang and Cheng Zhang |
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Vol. 14 (2021), No. 3, 861–880
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Abstract |
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We estimate the norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus , . We reduce getting correct bounds to counting lattice points in the intersection of some -transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain–Rudnick on -restriction estimates for rational hyperplanes. On , we prove the uniform restriction bounds for closed geodesics. On , we obtain explicit restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge. |
Keywords
eigenfunction estimates, restriction, geodesic
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Mathematical Subject Classification 2010
Primary: 11P21, 35P20, 58J50
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Milestones
Received: 27 February 2019
Revised: 24 September 2019
Accepted: 21 November 2019
Published: 18 May 2021
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Authors |
| Xiaoqi Huang | |
| Department of Mathematics Johns Hopkins University Baltimore, MD United States |
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| Cheng Zhang | |
| Department of Mathematics Johns Hopkins University Baltimore, MD United States |
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